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refactor: Change case of type and props

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jstoobysmith 2024-06-26 11:54:02 -04:00
parent 18b83f582e
commit f7a638d32e
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118
HepLean/AnomalyCancellation/MSSMNu/OrthogY3B3/ToSols.lean
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@ -38,14 +38,14 @@ namespace AnomalyFreePerp
/-- A condition for the quad line in the plane spanned by R, Y₃ and B₃ to sit in the cubic,
and for the cube line to sit in the quad. -/
def lineEqProp (R : MSSMACC.AnomalyFreePerp) : Prop := α₁ R = 0 ∧ α₂ R = 0 ∧ α₃ R = 0
def LineEqProp (R : MSSMACC.AnomalyFreePerp) : Prop := α₁ R = 0 ∧ α₂ R = 0 ∧ α₃ R = 0
instance (R : MSSMACC.AnomalyFreePerp) : Decidable (lineEqProp R) := by
instance (R : MSSMACC.AnomalyFreePerp) : Decidable (LineEqProp R) := by
apply And.decidable
/-- A condition on `Sols` which we will show in `linEqPropSol_iff_proj_linEqProp` that is equivalent
to the condition that the `proj` of the solution satisfies `lineEqProp`. -/
def lineEqPropSol (R : MSSMACC.Sols) : Prop :=
def LineEqPropSol (R : MSSMACC.Sols) : Prop :=
cubeTriLin R.val R.val Y₃.val * quadBiLin B₃.val R.val -
cubeTriLin R.val R.val B₃.val * quadBiLin Y₃.val R.val = 0
@ -54,8 +54,8 @@ equivalent to satisfying `lineEqPropSol`. -/
def lineEqCoeff (T : MSSMACC.Sols) : ℚ := dot Y₃.val B₃.val * α₃ (proj T.1.1)
lemma lineEqPropSol_iff_lineEqCoeff_zero (T : MSSMACC.Sols) :
lineEqPropSol T ↔ lineEqCoeff T = 0 := by
rw [lineEqPropSol, lineEqCoeff, α₃]
LineEqPropSol T ↔ lineEqCoeff T = 0 := by
rw [LineEqPropSol, lineEqCoeff, α₃]
simp only [Fin.isValue, Fin.reduceFinMk, mul_eq_zero, OfNat.ofNat_ne_zero,
false_or]
rw [cube_proj_proj_B₃, cube_proj_proj_Y₃, quad_B₃_proj, quad_Y₃_proj]
@ -66,8 +66,8 @@ lemma lineEqPropSol_iff_lineEqCoeff_zero (T : MSSMACC.Sols) :
simp
lemma linEqPropSol_iff_proj_linEqProp (R : MSSMACC.Sols) :
lineEqPropSol R ↔ lineEqProp (proj R.1.1) := by
rw [lineEqPropSol_iff_lineEqCoeff_zero, lineEqCoeff, lineEqProp]
LineEqPropSol R ↔ LineEqProp (proj R.1.1) := by
rw [lineEqPropSol_iff_lineEqCoeff_zero, lineEqCoeff, LineEqProp]
apply Iff.intro
intro h
rw [show dot Y₃.val B₃.val = 108 by rfl] at h
@ -81,15 +81,15 @@ lemma linEqPropSol_iff_proj_linEqProp (R : MSSMACC.Sols) :
/-- A condition which is satisfied if the plane spanned by `R`, `Y₃` and `B₃` lies
entirely in the quadratic surface. -/
def inQuadProp (R : MSSMACC.AnomalyFreePerp) : Prop :=
def InQuadProp (R : MSSMACC.AnomalyFreePerp) : Prop :=
quadBiLin R.val R.val = 0 ∧ quadBiLin Y₃.val R.val = 0 ∧ quadBiLin B₃.val R.val = 0
instance (R : MSSMACC.AnomalyFreePerp) : Decidable (inQuadProp R) := by
instance (R : MSSMACC.AnomalyFreePerp) : Decidable (InQuadProp R) := by
apply And.decidable
/-- A condition which is satisfied if the plane spanned by the solutions `R`, `Y₃` and `B₃`
lies entirely in the quadratic surface. -/
def inQuadSolProp (R : MSSMACC.Sols) : Prop :=
def InQuadSolProp (R : MSSMACC.Sols) : Prop :=
quadBiLin Y₃.val R.val = 0 ∧ quadBiLin B₃.val R.val = 0
/-- A rational which has two properties. It is zero for a solution `T` if and only if
@ -98,7 +98,7 @@ def quadCoeff (T : MSSMACC.Sols) : ℚ :=
2 * dot Y₃.val B₃.val ^ 2 *
(quadBiLin Y₃.val T.val ^ 2 + quadBiLin B₃.val T.val ^ 2)
lemma inQuadSolProp_iff_quadCoeff_zero (T : MSSMACC.Sols) : inQuadSolProp T ↔ quadCoeff T = 0 := by
lemma inQuadSolProp_iff_quadCoeff_zero (T : MSSMACC.Sols) : InQuadSolProp T ↔ quadCoeff T = 0 := by
apply Iff.intro
intro h
rw [quadCoeff, h.1, h.2]
@ -117,8 +117,8 @@ lemma inQuadSolProp_iff_quadCoeff_zero (T : MSSMACC.Sols) : inQuadSolProp T ↔
/-- The conditions `inQuadSolProp R` and `inQuadProp (proj R.1.1)` are equivalent. This is to be
expected since both `R` and `proj R.1.1` define the same plane with `Y₃` and `B₃`. -/
lemma inQuadSolProp_iff_proj_inQuadProp (R : MSSMACC.Sols) :
inQuadSolProp R ↔ inQuadProp (proj R.1.1) := by
rw [inQuadSolProp, inQuadProp]
InQuadSolProp R ↔ InQuadProp (proj R.1.1) := by
rw [InQuadSolProp, InQuadProp]
rw [quad_proj, quad_Y₃_proj, quad_B₃_proj]
apply Iff.intro
intro h
@ -133,17 +133,17 @@ lemma inQuadSolProp_iff_proj_inQuadProp (R : MSSMACC.Sols) :
/-- A condition which is satisfied if the plane spanned by `R`, `Y₃` and `B₃` lies
entirely in the cubic surface. -/
def inCubeProp (R : MSSMACC.AnomalyFreePerp) : Prop :=
def InCubeProp (R : MSSMACC.AnomalyFreePerp) : Prop :=
cubeTriLin R.val R.val R.val = 0 ∧ cubeTriLin R.val R.val B₃.val = 0 ∧
cubeTriLin R.val R.val Y₃.val = 0
instance (R : MSSMACC.AnomalyFreePerp) : Decidable (inCubeProp R) := by
instance (R : MSSMACC.AnomalyFreePerp) : Decidable (InCubeProp R) := by
apply And.decidable
/-- A condition which is satisfied if the plane spanned by the solutions `R`, `Y₃` and `B₃`
lies entirely in the cubic surface. -/
def inCubeSolProp (R : MSSMACC.Sols) : Prop :=
def InCubeSolProp (R : MSSMACC.Sols) : Prop :=
cubeTriLin R.val R.val B₃.val = 0 ∧ cubeTriLin R.val R.val Y₃.val = 0
/-- A rational which has two properties. It is zero for a solution `T` if and only if
@ -153,7 +153,7 @@ def cubicCoeff (T : MSSMACC.Sols) : ℚ :=
cubeTriLin T.val T.val B₃.val ^ 2 )
lemma inCubeSolProp_iff_cubicCoeff_zero (T : MSSMACC.Sols) :
inCubeSolProp T ↔ cubicCoeff T = 0 := by
InCubeSolProp T ↔ cubicCoeff T = 0 := by
apply Iff.intro
intro h
rw [cubicCoeff, h.1, h.2]
@ -170,8 +170,8 @@ lemma inCubeSolProp_iff_cubicCoeff_zero (T : MSSMACC.Sols) :
exact h.symm
lemma inCubeSolProp_iff_proj_inCubeProp (R : MSSMACC.Sols) :
inCubeSolProp R ↔ inCubeProp (proj R.1.1) := by
rw [inCubeSolProp, inCubeProp]
InCubeSolProp R ↔ InCubeProp (proj R.1.1) := by
rw [InCubeSolProp, InCubeProp]
rw [cube_proj, cube_proj_proj_Y₃, cube_proj_proj_B₃]
apply Iff.intro
intro h
@ -186,30 +186,30 @@ lemma inCubeSolProp_iff_proj_inCubeProp (R : MSSMACC.Sols) :
/-- Those charge assignments perpendicular to `Y₃` and `B₃` which satisfy the condition
`lineEqProp`. -/
def inLineEq : Type := {R : MSSMACC.AnomalyFreePerp // lineEqProp R}
def InLineEq : Type := {R : MSSMACC.AnomalyFreePerp // LineEqProp R}
/-- Those charge assignments perpendicular to `Y₃` and `B₃` which satisfy the condition
`lineEqProp` and `inQuadProp`. -/
def inQuad : Type := {R : inLineEq // inQuadProp R.val}
def InQuad : Type := {R : InLineEq // InQuadProp R.val}
/-- Those charge assignments perpendicular to `Y₃` and `B₃` which satisfy the condition
`lineEqProp`, `inQuadProp` and `inCubeProp`. -/
def inQuadCube : Type := {R : inQuad // inCubeProp R.val.val}
def InQuadCube : Type := {R : InQuad // InCubeProp R.val.val}
/-- Those solutions which do not satisfy the condition `lineEqPropSol`. -/
def notInLineEqSol : Type := {R : MSSMACC.Sols // ¬ lineEqPropSol R}
def NotInLineEqSol : Type := {R : MSSMACC.Sols // ¬ LineEqPropSol R}
/-- Those solutions which satisfy the condition `lineEqPropSol` by not `inQuadSolProp`. -/
def inLineEqSol : Type := {R : MSSMACC.Sols // lineEqPropSol R ∧ ¬ inQuadSolProp R}
def InLineEqSol : Type := {R : MSSMACC.Sols // LineEqPropSol R ∧ ¬ InQuadSolProp R}
/-- Those solutions which satisfy the condition `lineEqPropSol` and `inQuadSolProp` but
not `inCubeSolProp`. -/
def inQuadSol : Type := {R : MSSMACC.Sols // lineEqPropSol R ∧ inQuadSolProp R ∧ ¬ inCubeSolProp R}
def InQuadSol : Type := {R : MSSMACC.Sols // LineEqPropSol R ∧ InQuadSolProp R ∧ ¬ InCubeSolProp R}
/-- Those solutions which satisfy the condition all the conditions `lineEqPropSol`, `inQuadSolProp`
and `inCubeSolProp`. -/
def inQuadCubeSol : Type :=
{R : MSSMACC.Sols // lineEqPropSol R ∧ inQuadSolProp R ∧ inCubeSolProp R}
def InQuadCubeSol : Type :=
{R : MSSMACC.Sols // LineEqPropSol R ∧ InQuadSolProp R ∧ InCubeSolProp R}
/-- Given a `R` perpendicular to `Y₃` and `B₃` a quadratic solution. -/
def toSolNSQuad (R : MSSMACC.AnomalyFreePerp) : MSSMACC.QuadSols :=
@ -244,7 +244,7 @@ def toSolNS : MSSMACC.AnomalyFreePerp × ℚ × ℚ × ℚ → MSSMACC.Sols := f
def toSolNSProj (T : MSSMACC.Sols) : MSSMACC.AnomalyFreePerp × ℚ × ℚ × ℚ :=
(proj T.1.1, (lineEqCoeff T)⁻¹, 0, 0)
lemma toSolNS_proj (T : notInLineEqSol) : toSolNS (toSolNSProj T.val) = T.val := by
lemma toSolNS_proj (T : NotInLineEqSol) : toSolNS (toSolNSProj T.val) = T.val := by
apply ACCSystem.Sols.ext
rw [toSolNS, toSolNSProj]
change (lineEqCoeff T.val)⁻¹ • (toSolNSQuad _).1.1 = _
@ -263,7 +263,7 @@ lemma toSolNS_proj (T : notInLineEqSol) : toSolNS (toSolNSProj T.val) = T.val :=
simp
/-- Given a element of `inLineEq × ℚ × ℚ × ℚ`, a solution to the ACCs. -/
def inLineEqToSol : inLineEq × ℚ × ℚ × ℚ → MSSMACC.Sols := fun (R, c₁, c₂, c₃) =>
def inLineEqToSol : InLineEq × ℚ × ℚ × ℚ → MSSMACC.Sols := fun (R, c₁, c₂, c₃) =>
AnomalyFreeMk'' (lineQuad R.val c₁ c₂ c₃)
(by
rw [lineQuad_cube]
@ -271,7 +271,7 @@ def inLineEqToSol : inLineEq × ℚ × ℚ × ℚ → MSSMACC.Sols := fun (R, c
simp)
/-- On elements of `inLineEqSol` a right-inverse to `inLineEqSol`. -/
def inLineEqProj (T : inLineEqSol) : inLineEq × ℚ × ℚ × ℚ :=
def inLineEqProj (T : InLineEqSol) : InLineEq × ℚ × ℚ × ℚ :=
(⟨proj T.val.1.1, (linEqPropSol_iff_proj_linEqProp T.val).mp T.prop.1 ⟩,
(quadCoeff T.val)⁻¹ * quadBiLin B₃.val T.val.val,
(quadCoeff T.val)⁻¹ * (- quadBiLin Y₃.val T.val.val),
@ -279,14 +279,14 @@ def inLineEqProj (T : inLineEqSol) : inLineEq × ℚ × ℚ × ℚ :=
quadBiLin B₃.val T.val.val * (dot B₃.val T.val.val - dot Y₃.val T.val.val)
- quadBiLin Y₃.val T.val.val * (dot Y₃.val T.val.val - 2 * dot B₃.val T.val.val)))
lemma inLineEqTo_smul (R : inLineEq) (c₁ c₂ c₃ d : ℚ) :
lemma inLineEqTo_smul (R : InLineEq) (c₁ c₂ c₃ d : ℚ) :
inLineEqToSol (R, (d * c₁), (d * c₂), (d * c₃)) = d • inLineEqToSol (R, c₁, c₂, c₃) := by
apply ACCSystem.Sols.ext
change (lineQuad _ _ _ _).val = _
rw [lineQuad_smul]
rfl
lemma inLineEqToSol_proj (T : inLineEqSol) : inLineEqToSol (inLineEqProj T) = T.val := by
lemma inLineEqToSol_proj (T : InLineEqSol) : inLineEqToSol (inLineEqProj T) = T.val := by
rw [inLineEqProj, inLineEqTo_smul]
apply ACCSystem.Sols.ext
change _ • (lineQuad _ _ _ _).val = _
@ -306,7 +306,7 @@ lemma inLineEqToSol_proj (T : inLineEqSol) : inLineEqToSol (inLineEqProj T) = T.
simp
/-- Given a element of `inQuad × ℚ × ℚ × ℚ`, a solution to the ACCs. -/
def inQuadToSol : inQuad × ℚ × ℚ × ℚ → MSSMACC.Sols := fun (R, a₁, a₂, a₃) =>
def inQuadToSol : InQuad × ℚ × ℚ × ℚ → MSSMACC.Sols := fun (R, a₁, a₂, a₃) =>
AnomalyFreeMk' (lineCube R.val.val a₁ a₂ a₃)
(by
erw [planeY₃B₃_quad]
@ -314,7 +314,7 @@ def inQuadToSol : inQuad × ℚ × ℚ × ℚ → MSSMACC.Sols := fun (R, a₁,
simp)
(lineCube_cube R.val.val a₁ a₂ a₃)
lemma inQuadToSol_smul (R : inQuad) (c₁ c₂ c₃ d : ℚ) :
lemma inQuadToSol_smul (R : InQuad) (c₁ c₂ c₃ d : ℚ) :
inQuadToSol (R, (d * c₁), (d * c₂), (d * c₃)) = d • inQuadToSol (R, c₁, c₂, c₃) := by
apply ACCSystem.Sols.ext
change (lineCube _ _ _ _).val = _
@ -322,7 +322,7 @@ lemma inQuadToSol_smul (R : inQuad) (c₁ c₂ c₃ d : ℚ) :
rfl
/-- On elements of `inQuadSol` a right-inverse to `inQuadToSol`. -/
def inQuadProj (T : inQuadSol) : inQuad × ℚ × ℚ × ℚ :=
def inQuadProj (T : InQuadSol) : InQuad × ℚ × ℚ × ℚ :=
(⟨⟨proj T.val.1.1, (linEqPropSol_iff_proj_linEqProp T.val).mp T.prop.1 ⟩,
(inQuadSolProp_iff_proj_inQuadProp T.val).mp T.prop.2.1⟩,
(cubicCoeff T.val)⁻¹ * (cubeTriLin T.val.val T.val.val B₃.val),
@ -332,7 +332,7 @@ def inQuadProj (T : inQuadSol) : inQuad × ℚ × ℚ × ℚ :=
- cubeTriLin T.val.val T.val.val Y₃.val
* (dot Y₃.val T.val.val - 2 * dot B₃.val T.val.val)))
lemma inQuadToSol_proj (T : inQuadSol) : inQuadToSol (inQuadProj T) = T.val := by
lemma inQuadToSol_proj (T : InQuadSol) : inQuadToSol (inQuadProj T) = T.val := by
rw [inQuadProj, inQuadToSol_smul]
apply ACCSystem.Sols.ext
change _ • (planeY₃B₃ _ _ _ _).val = _
@ -352,7 +352,7 @@ lemma inQuadToSol_proj (T : inQuadSol) : inQuadToSol (inQuadProj T) = T.val := b
simp
/-- Given a element of `inQuadCube × ℚ × ℚ × ℚ`, a solution to the ACCs. -/
def inQuadCubeToSol : inQuadCube × ℚ × ℚ × ℚ → MSSMACC.Sols := fun (R, b₁, b₂, b₃) =>
def inQuadCubeToSol : InQuadCube × ℚ × ℚ × ℚ → MSSMACC.Sols := fun (R, b₁, b₂, b₃) =>
AnomalyFreeMk' (planeY₃B₃ R.val.val.val b₁ b₂ b₃)
(by
rw [planeY₃B₃_quad]
@ -363,7 +363,7 @@ def inQuadCubeToSol : inQuadCube × ℚ × ℚ × ℚ → MSSMACC.Sols := fun (R
rw [R.prop.1, R.prop.2.1, R.prop.2.2]
simp)
lemma inQuadCubeToSol_smul (R : inQuadCube) (c₁ c₂ c₃ d : ℚ) :
lemma inQuadCubeToSol_smul (R : InQuadCube) (c₁ c₂ c₃ d : ℚ) :
inQuadCubeToSol (R, (d * c₁), (d * c₂), (d * c₃)) = d • inQuadCubeToSol (R, c₁, c₂, c₃):= by
apply ACCSystem.Sols.ext
change (planeY₃B₃ _ _ _ _).val = _
@ -371,7 +371,7 @@ lemma inQuadCubeToSol_smul (R : inQuadCube) (c₁ c₂ c₃ d : ℚ) :
rfl
/-- On elements of `inQuadCubeSol` a right-inverse to `inQuadCubeToSol`. -/
def inQuadCubeProj (T : inQuadCubeSol) : inQuadCube × ℚ × ℚ × ℚ :=
def inQuadCubeProj (T : InQuadCubeSol) : InQuadCube × ℚ × ℚ × ℚ :=
(⟨⟨⟨proj T.val.1.1, (linEqPropSol_iff_proj_linEqProp T.val).mp T.prop.1 ⟩,
(inQuadSolProp_iff_proj_inQuadProp T.val).mp T.prop.2.1⟩,
(inCubeSolProp_iff_proj_inCubeProp T.val).mp T.prop.2.2⟩,
@ -379,7 +379,7 @@ def inQuadCubeProj (T : inQuadCubeSol) : inQuadCube × ℚ × ℚ × ℚ :=
(dot Y₃.val B₃.val)⁻¹ * (2 * dot B₃.val T.val.val - dot Y₃.val T.val.val),
(dot Y₃.val B₃.val)⁻¹ * 1)
lemma inQuadCubeToSol_proj (T : inQuadCubeSol) :
lemma inQuadCubeToSol_proj (T : InQuadCubeSol) :
inQuadCubeToSol (inQuadCubeProj T) = T.val := by
rw [inQuadCubeProj, inQuadCubeToSol_smul]
apply ACCSystem.Sols.ext
@ -396,64 +396,64 @@ lemma inQuadCubeToSol_proj (T : inQuadCubeSol) :
/-- Given an element of `MSSMACC.AnomalyFreePerp × ℚ × ℚ × ℚ` a solution. We will
show that this map is a surjection. -/
def toSol : MSSMACC.AnomalyFreePerp × ℚ × ℚ × ℚ → MSSMACC.Sols := fun (R, a, b, c) =>
if h₃ : lineEqProp R ∧ inQuadProp R ∧ inCubeProp R then
if h₃ : LineEqProp R ∧ InQuadProp R ∧ InCubeProp R then
inQuadCubeToSol (⟨⟨⟨R, h₃.1⟩, h₃.2.1⟩, h₃.2.2⟩, a, b, c)
else
if h₂ : lineEqProp R ∧ inQuadProp R then
if h₂ : LineEqProp R ∧ InQuadProp R then
inQuadToSol (⟨⟨R, h₂.1⟩, h₂.2⟩, a, b, c)
else
if h₁ : lineEqProp R then
if h₁ : LineEqProp R then
inLineEqToSol (⟨R, h₁⟩, a, b, c)
else
toSolNS ⟨R, a, b, c⟩
lemma toSol_toSolNSProj (T : notInLineEqSol) :
lemma toSol_toSolNSProj (T : NotInLineEqSol) :
∃ X, toSol X = T.val := by
use toSolNSProj T.val
have h1 : ¬ lineEqProp (toSolNSProj T.val).1 :=
have h1 : ¬ LineEqProp (toSolNSProj T.val).1 :=
(linEqPropSol_iff_proj_linEqProp T.val).mpr.mt T.prop
rw [toSol]
simp_all
exact toSolNS_proj T
lemma toSol_inLineEq (T : inLineEqSol) : ∃ X, toSol X = T.val := by
lemma toSol_inLineEq (T : InLineEqSol) : ∃ X, toSol X = T.val := by
let X := inLineEqProj T
use ⟨X.1.val, X.2.1, X.2.2⟩
have : ¬ inQuadProp X.1.val := (inQuadSolProp_iff_proj_inQuadProp T.val).mpr.mt T.prop.2
have : lineEqProp X.1.val := (linEqPropSol_iff_proj_linEqProp T.val).mp T.prop.1
have : ¬ InQuadProp X.1.val := (inQuadSolProp_iff_proj_inQuadProp T.val).mpr.mt T.prop.2
have : LineEqProp X.1.val := (linEqPropSol_iff_proj_linEqProp T.val).mp T.prop.1
rw [toSol]
simp_all
exact inLineEqToSol_proj T
lemma toSol_inQuad (T : inQuadSol) : ∃ X, toSol X = T.val := by
lemma toSol_inQuad (T : InQuadSol) : ∃ X, toSol X = T.val := by
let X := inQuadProj T
use ⟨X.1.val.val, X.2.1, X.2.2⟩
have : ¬ inCubeProp X.1.val.val := (inCubeSolProp_iff_proj_inCubeProp T.val).mpr.mt T.prop.2.2
have : inQuadProp X.1.val.val := (inQuadSolProp_iff_proj_inQuadProp T.val).mp T.prop.2.1
have : lineEqProp X.1.val.val := (linEqPropSol_iff_proj_linEqProp T.val).mp T.prop.1
have : ¬ InCubeProp X.1.val.val := (inCubeSolProp_iff_proj_inCubeProp T.val).mpr.mt T.prop.2.2
have : InQuadProp X.1.val.val := (inQuadSolProp_iff_proj_inQuadProp T.val).mp T.prop.2.1
have : LineEqProp X.1.val.val := (linEqPropSol_iff_proj_linEqProp T.val).mp T.prop.1
rw [toSol]
simp_all
exact inQuadToSol_proj T
lemma toSol_inQuadCube (T : inQuadCubeSol) : ∃ X, toSol X = T.val := by
lemma toSol_inQuadCube (T : InQuadCubeSol) : ∃ X, toSol X = T.val := by
let X := inQuadCubeProj T
use ⟨X.1.val.val.val, X.2.1, X.2.2⟩
have : inCubeProp X.1.val.val.val := (inCubeSolProp_iff_proj_inCubeProp T.val).mp T.prop.2.2
have : inQuadProp X.1.val.val.val := (inQuadSolProp_iff_proj_inQuadProp T.val).mp T.prop.2.1
have : lineEqProp X.1.val.val.val := (linEqPropSol_iff_proj_linEqProp T.val).mp T.prop.1
have : InCubeProp X.1.val.val.val := (inCubeSolProp_iff_proj_inCubeProp T.val).mp T.prop.2.2
have : InQuadProp X.1.val.val.val := (inQuadSolProp_iff_proj_inQuadProp T.val).mp T.prop.2.1
have : LineEqProp X.1.val.val.val := (linEqPropSol_iff_proj_linEqProp T.val).mp T.prop.1
rw [toSol]
simp_all
exact inQuadCubeToSol_proj T
theorem toSol_surjective : Function.Surjective toSol := by
intro T
by_cases h₁ : ¬ lineEqPropSol T
by_cases h₁ : ¬ LineEqPropSol T
exact toSol_toSolNSProj ⟨T, h₁⟩
simp at h₁
by_cases h₂ : ¬ inQuadSolProp T
by_cases h₂ : ¬ InQuadSolProp T
exact toSol_inLineEq ⟨T, And.intro h₁ h₂⟩
simp at h₂
by_cases h₃ : ¬ inCubeSolProp T
by_cases h₃ : ¬ InCubeSolProp T
exact toSol_inQuad ⟨T, And.intro h₁ (And.intro h₂ h₃)⟩
simp at h₃
exact toSol_inQuadCube ⟨T, And.intro h₁ (And.intro h₂ h₃)⟩